Banach algebra mappings preserving the invertibility of linear pencils

Abstract

Let $A$ and $B$ be complex unital Banach algebras, and let $\phi ,\psi :A\to B$ be surjective mappings. If $A$ is semisimple with an essential socle and $\phi$ and $\psi$ preserves the invertibility of linear pencils in both directions, that is, for any $x,y\in A$ and $\lambda \in \mathbb{C}$, $\lambda x+y$ is invertible in $A$ if and only if $\lambda \phi (x)+\psi (y)$ is invertible in $B$, then we show that there exists an invertible element $u$ in $B$ and a Jordan isomorphism $J:A\to B$ such that $\phi (x)=\psi (x)=uJ(x)$ for all $x\in A$.

12 pages